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Abstract and Applied Analysis
Volume 2010 (2010), Article ID 123094, 14 pages
http://dx.doi.org/10.1155/2010/123094
Research Article

Some Nonunique Fixed Point Theorems of Ćirić Type on Cone Metric Spaces

Department of Mathematics, Atılım University, 06836 Ankara, Turkey

Received 18 January 2010; Revised 6 March 2010; Accepted 15 April 2010

Academic Editor: Dumitru Baleanu

Copyright © 2010 Erdal Karapınar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Some results of (Ćirić, 1974) on a nonunique fixed point theorem on the class of metric spaces are extended to the class of cone metric spaces. Namely, nonunique fixed point theorem is proved in orbitally complete cone metric spaces under the assumption that the cone is strongly minihedral. Regarding the scalar weight of cone metric, we are able to remove the assumption of strongly minihedral.

1. Introduction and Preliminaries

In 1980, Rzepecki [1] introduced a generalized metric on a set in a way that where is a Banach space and is a normal cone in with partial order . In that paper, the author generalized the fixed point theorems of Maia type [2].

In 1987, Lin [3] considered the notion of -metric spaces by replacing real numbers with cone in the metric function, that is, . In that manuscript, some results of Khan and Imdad [4] on fixed point theorems were considered for -metric spaces. Without mentioning the papers of Lin and Rzepecki, in 2007, Huang and Zhang [5] announced the notion of cone metric spaces (CMSs) by replacing real numbers with an ordering Banach space. In that paper, they also discussed some properties of convergence of sequences and proved the fixed point theorems of contractive mapping for cone metric spaces.

Recently, many results on fixed point theory have been extended to cone metric spaces (see, e.g., [513]).

Ćirić type nonunique fixed point theorems were considered by many authors (see, e.g., [1420]). In this paper, some of the known results (see, e.g., [2, 14, 15]) are extended to cone metric spaces.

Throughout this paper stands for a real Banach space. Let always be a closed nonempty subset of . is called cone if for all and non-negative real numbers where and .

For a given cone , one can define a partial ordering (denoted by or ) with respect to by if and only if . The notation indicates that and while will show , where denotes the interior of . From now on, it is assumed that

The cone is called normal if there is a number for which holds for all . The least positive integer , satisfying this equation, is called the normal constant of . The cone is said to be regular if every increasing sequence which is bounded from above is convergent, that is, if is a sequence such that for some , then there is such that .

Lemma 1.1. (i) Every regular cone is normal.
(ii) For each , there is a normal cone with normal constant .
(iii) The cone is regular if every decreasing sequence which is bounded from below is convergent.

Proof of (i) and (ii) are given in [6] and the last one follows from definition.

Definition 1.2. Let be a nonempty set. Suppose that the mapping satisfies for all , if and only if , , for all , for all then is called cone metric on , and the pair is called a cone metric space (CMS).

Example 1.3. Let , and . Define by , where are positive constants. Then is a CMS. Note that the cone is normal with the normal constant

Definition 1.4. Let be a CMS, , and a sequence in . Then (i)  converges to whenever for every with there is a natural number , such that for all . It is denoted by or . (ii)  is a Cauchy sequence whenever for every with there is a natural number , such that for all . (iii) is a complete cone metric space if every Cauchy sequence is convergent.

Lemma 1.5 (see [5]). Let be a CMS, a normal cone with normal constant , and a sequence in . Then, (i)the sequence converges to if and only if (or equivalently ),(ii)the sequence is Cauchy if and only if (or equivalently ),(iii)the sequence converges to and the sequence converges to , then .

Lemma 1.6 (see [8]). Let be a CMS over a cone in . Then() and .()If , then there exists such that implies that .()For any given and there exists such that .()If are sequences in such that , , and , then .

Definition 1.7 (see [21]). is called minihedral cone if exists for all and strongly minihedral if every subset of which is bounded from above has a supremum. (equivalently, if every subset of which is bounded from below has an infimum.)

Lemma 1.8. (i) Every strongly minihedral normal (not necessarily closed) cone is regular.
(ii) Every strongly minihedral (closed) cone is normal.

The proof of (i) is straightforward, and for (ii) see, for example, [22].

Example 1.9. Let with the supremum norm and Then is a cone with normal constant which is not regular. This is clear, since the sequence is monotonically decreasing but not uniformly convergent to . This cone, by Lemma 1.8, is not strongly minihedral. However, it is easy to see that the cone mentioned in Example 1.3 is strongly minihedral.

2. Non unique Fixed Points on Cone Metric Spaces

Definition 2.1. A mapping on CMS is said to be orbitally continuous if implies that . A CMS is called orbitally complete if every Cauchy sequence of the form , converges in .

Remark 2.2. It is clear that orbital continuity of implies orbital continuity of for any .

Theorem 2.3. Let be an orbitally continuous mapping on CMS over strongly minihedral normal cone . Suppose that CMS is orbitally complete and that satisfies the condition for all and for some , where . Then, for each , the iterated sequence converges to a fixed point of .

Proof. Fix . For set and recursively . It is clear that the sequence is Cauchy when the equation holds for some . Consider the case for all . By replacing and with and , respectively, in (2.1), one can get where . Since , the case yields contradiction. Thus, . Recursively, one can observe that By using the triangle inequality, for any , one can get Let . Choose a natural number such that for all . Thus, for any , for all . So is a Cauchy sequence in . Since is orbitally complete, there is some such that . Regarding the orbital continuity of , , that is, is a fixed point of .

A point is said to be a periodic point of a function of period if , where and is defined recursively by .

Theorem 2.4. Let be an orbitally continuous mapping on orbitally complete CMS over strongly minihedral normal cone and . Suppose that there exists a point such that for some and that satisfies the condition for all and for some , where . Then, has a periodic point.

Proof. Set . By assumption of theorem Set and let such that which is equivalent to saying that .
Suppose that . By replacing in (2.5), one can get where . There are two cases. Consider the first case, , which is a contraction by, regarding . Thus, one has .
As in the proof of Theorem 2.3, one can consider the iterative sequence and observe that for some .
Suppose that . It is equivalent to saying that for each , the condition Taking account of and applying into (2.5), one can get where .
Recall that and say that . Then, is observed. Regarding (2.7), and also . Thus, and hence, (2.8) turns into Recursively, one can get Continuing in this way, for each , one can obtain Thus, for the recursive sequence where , By using the triangle inequality, for any , one can get Let . Choose a natural number such that   for all . Thus, for any , for all . So is a Cauchy sequence in . Since is orbitally complete, there is some such that . Regarding Remark 2.2, the orbital continuity of implies that that is, is a periodic point of .

Theorem 2.5. Let be an orbitally continuous mapping on CMS over strongly minihedral normal cone . Suppose that satisfies the condition for all where . Suppose that the sequence has a cluster point , for some . Then, is a fixed point of .

Proof. Suppose that for some , then for all . It is clear that is a required point.
Suppose that for all . Since has a cluster point , one can write . By replacing and with and , respectively, in (2.16), where ),)) lies in  , . The case is impossible. Thus, (2.17) is equivalent to . It shows that is decreasing. Since the cone is strongly minihedral, then by Lemma 1.1 (iii) and Lemma 1.8 (i), is convergent. Due to Lemma 1.5, and orbital continuity, By and (2.19), Regarding , , and (2.20), Assume that , that is, . So, one can replace and with and , respectively, in(2.16) where .
It yields that . But it contradicts (2.21). Thus, .

3. Non unique Fixed Points on Scalar Weighted Cone Metric Spaces

Definition 3.1. Let be a CMS. The scalar weight of the cone metric is defined by .

Notice that for normal cone with the normal constant , the scalar weight of the cone metric behaves as a metric on . In the following theorems normal constant has no restriction.

Theorem 3.2. Let be an orbitally continuous mapping on orbitally complete CMS over normal cone with normal constant . Suppose that satisfies the condition for all and for some . Then, for each , the iterated sequence converges to a fixed point of .

Proof. Fix . For set and recursively . It is clear that the sequence is Cauchy when hold for some . Consider the case for all . By replacing and with and , respectively, in (3.1), one can get
Since , the case yields contradiction. Thus, . Recursively, one can observe that By using the triangle inequality, for any , one can get By routine calculation, one can obtain that is a Cauchy sequence in . Since is orbitally complete, there is some such that Regarding the orbital continuity of , that is, is a fixed point of .

Theorem 3.3. Let be an orbitally continuous mapping on orbitally complete CMS over normal cone with normal constant and . Suppose that there exists a point such that for some and that satisfies the condition for all and for some . Then, has a periodic point.

Proof. Set . By assumption of the theorem Let and such that .
Suppose that , that is, . By replacing in (3.7), one can get The case implies a contraction due to the fact that . Thus, .
As in the proof of Theorem 3.2, one can consider the iterative sequence and observe that for some .
Suppose that . It is equivalent to saying that the condition holds for each . Then, from and (3.7) it follows that Considering , say , one has . Regarding (3.9), and . Thus, and hence Recursively, one can get Continuing in this way, for each , one can obtain Thus, for the recursive sequence where , By using the triangle inequality and regarding the normality of the cone, for any , one can get
Let . Choose a natural number such that for all . Thus, for any ,   for all . So is a Cauchy sequence in . Since is orbitally complete, there is some such that . Regarding Remark 2.2, the orbital continuity of implies that that is, is a periodic point of .

Theorem 3.4. Let be an orbitally continuous mapping on CMS over normal cone with normal constant . Suppose that satisfies the condition for all . If the sequence has a cluster point , for some , then is a fixed point of .

The proof of Theorem 3.4 is omitted by regarding the analogy with the proof of Theorem 2.5. In the proof of Theorem 2.5, to conclude that the decreasing sequence (2.18) is convergent, we need to use the assumption of strong minihedrality of the cone . Since we use the scalar weight of cone metric in the proof of Theorem 3.4, we can conclude that the corresponding decreasing sequence of (2.18) is convergent without the assumption of strong minihedrality of the cone .

Theorem 3.5. Let be an orbitally continuous mapping on orbitally complete CMS over normal cone with normal constant and . Suppose that satisfies the condition for all . If for some , the sequence has a cluster point of , then is a periodic point of .

Proof. Set , that is, for any there exists such that for all . Hence, by triangle inequality and normality of the cone it yields that Define a set which is nonempty by assumption of the theorem. Let . Consider two cases. Suppose for some . Then, and the assertion of theorem follows.
Suppose that for all . Let be such that
If , then replacing and with and , respectively, in (3.19) one can obtain that Since the case is impossible, (3.22) turns into , that is, the sequence is decreasing for . Thus, by routine calculation, one can conclude that .
Assume that , that is, for every , By orbital continuity of , , and by (3.23), one can get for every
Regarding (3.19) under the assumption one can obtain Thus, due to (3.23), .
By continuing this process, it yields that
Hence, the sequence is decreasing and thus is convergent. Notice that the subsequences and are convergent to and , respectively. By orbital continuity of and , one can get
One can conclude that   from (3.26) and (3.27). If , then . Thus, the desired result is obtained. Suppose that . Applying (3.19), Taking account of (3.24), (3.28) yields that which contradicts (3.27). Thus, , and so .

Theorem 3.6. Let be an orbitally continuous mapping on orbitally complete CMS over normal cone with normal constant . Suppose that satisfies the condition for all and for some . Then, for each , the iterated sequence converges to a fixed point of .

Proof. As in the proof of Theorem 3.2, fix and define the sequence in the following way. For set and recursively . It is clear that the sequence is Cauchy when hold for some . Consider the case for all . By replacing and with and , respectively, in (3.29), one can get Since , the case yields contradiction. Thus, one gets Recursively, one can observe that By routine calculation as in the proof of Theorem 3.2, one can show that has a fixed point.

Theorem 3.7. Let be a nonempty set endowed in two cone metrics , and let be a mapping of into itself. Suppose that (i) is orbitally complete space with respect to , (ii) for all , (iii) is orbitally continuous with respect to , (iv) satisfies for all , where . Then has a fixed point in .

Proof. As in the proof of Theorem 3.2, fix and define the sequence in the following way. For set and recursively . Replacing with , respectively, in (3.33), one can get Since the case (3.34) is equivalent to . Recursively one can obtain Regarding the triangle inequality and the normality of the cone, (3.35) implies that for any . Taking account of assumption (ii) of the theorem, one can get Thus, is a Cauchy sequence with respect to . Since is orbitally complete, there exists such that . From orbital continuity of , one can get the desired result, that is,
Remark 3.8. Theorem 3.6 can be restated by replacing (3.29) with Note also that, Theorem 3.7 remains valid by replacing (3.33) with

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